plot::Integral – numerical approximation of an integral

plot::Integral(f, IntMethod = m) visualizes the approximation of the integral of the function f using the numerical quadrature method m. Riemann sums, the trapezoidal rule, and the Simpson rule are available.

plot::Integral(f, n, IntMethod = m) uses n subintervals to approximate the integral.

→ Examples

Call:

plot::Integral(f, <n>, <IntMethod = m>, <a = amin .. amax>, Options)

Parameters:

f:	the integrand: an object of type plot::Function2d. f is equivalent to the attribute Function1.
n:	the number of subintervals (a positive integer) or a list of real numbers representing nodes of the integration variable. n is equivalent to the attribute Nodes.

Options:

IntMethod = m:

the quadrature method; see IntMethod

Related Functions:

plot, plot::copy

Details:

plot::Integral visualizes approximations of the integral over a function f. The attribute IntMethod determines the numerical method. Riemann sums, the trapezoidal rule, or the Simpson rule are available. See the help page of IntMethod for further details. Cf. example 1.
plot::Integral does not plot the function graph of the integrand. If the integrand is to be plotted, too, f has to be passed to the plot command together with the approximation object of type plot::Integral.
If no quadrature method is specified by IntMethod = m, plot::Integral just hatches the area between the function f and the x-axis.
Several plot::Integral objects can be plotted together to illustrate the difference between various quadrature methods. The order of the objects in the plot command determines the object in front.
The plot contains a text object providing information about the quadrature method, the value of the approximation, the exact value of the integral, the quadrature error, and the number of nodes. See the help page of the attribute ShowInfo for further details.

Example 1

If a single plot::Function2d object is given without specifying an approximation method, plot::Integral just hatches the area between the function graph and the x-axis:

f := plot::Function2d(cos(x), x = -PI..PI):
plot(plot::Integral(f), f)

MuPAD graphics

Note that plot::Integral requires an object of type plot::Function2d, not just a function expression:

plot::Integral(sin(x))

Error: Expecting a 'plot::Function2d' object as first argument. [plot::Integral::new]

If an approximation method is specified, the numerical quadrature value computed by this method is displayed:

plot(plot::Integral(f, IntMethod = RiemannLower))

MuPAD graphics

The number of quadrature intervals can be set by passing a second argument n or by specifying Nodes = n:

plot(plot::Integral(f, 20, IntMethod = RiemannLower))

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To see the integrand in the plot, the function object must be passed together with the approximation object. The order determines which object is in front:

plot(plot::Integral(f, IntMethod = RiemannLower), f)

MuPAD graphics

delete f:

Example 2

The displayed information can be configured by the user:

f := plot::Function2d(x^2, x = -5..5, Color = RGB::DarkGrey):
plot(plot::Integral(f, IntMethod = RiemannLower,
       ShowInfo = [1, IntMethod = "Riemann lower sum",
                   Integral = "Exact value",
                   2, Error = "Difference"]), f)

MuPAD graphics

delete f:

Example 3

One may combine several approximation objects, e.g., lower and upper sum:

f := plot::Function2d(x^2, x = -5..5):
plot(plot::Integral(f, IntMethod = RiemannUpper,
                       Color = RGB::Blue),
     plot::Integral(f, IntMethod = RiemannLower,
                       Color = RGB::LightYellow),
     f)

MuPAD graphics

The automatically placed information texts overlap. To correct this, the option ShowInfo must be used. In the text of the upper sum, one additional empty line is inserted. Apart from this, both objects use the default value, therefore there is not need to specify ShowInfo in the second object:

plot(plot::Integral(f, IntMethod = RiemannUpper,
                       ShowInfo = [IntMethod, "", Integral]),
     plot::Integral(f, IntMethod = RiemannLower,
                       Color = RGB::LightYellow),
     f)

MuPAD graphics

The info text can be positioned explicitly:

plot(plot::Integral(f, IntMethod = RiemannUpper,
                       ShowInfo = [IntMethod, Integral,
                                   Position = [-5, -1]],
                       VerticalAlignment = Top),
     plot::Integral(f, IntMethod = RiemannLower, Color = RGB::Yellow,
                       ShowInfo = [IntMethod,
                                   Position = [0, -1]],
                       VerticalAlignment = Top),
     f)

MuPAD graphics

delete f:

Example 4

plot::Integral can be animated:

f := plot::Function2d(sin(a*x), x = 0..PI, a = 1..5):
plot(plot::Integral(f, 50, IntMethod = RiemannMiddle), f)

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Increasing the number of nodes decreases the quadrature error:

f := plot::Function2d(sin(x), x = 0..PI):
plot(plot::Integral(f, N, N = 10..50, IntMethod = RiemannLower), f)

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The function and the number of nodes can be animated simultaneously:

f := plot::Function2d(sin(a*x), x = 0..PI, a = 1..5):
plot(plot::Integral(f, N, N = 10..50, IntMethod = RiemannLower), f)

MuPAD graphics

delete f:

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